Let $J$ denote the image of $J$-homomorphism spectrum and let $j$ denote its connective cover. I am interested in knowing the cohomology of $j$ i.e. $$ [j, HZ/p]_*$$ as a module over Steenrod algebra. Or dually the homology of $j$ as a comodule over the dual Steenrod algebra. Is there a nice description?

A reference will be appreciated as well.

  • 3
    $\begingroup$ If I'm not mistaken this is the subject of Watanabe's paper "On the spectrum representing algebraic K-theory for a finite field", see projecteuclid.org/euclid.ojm/1200778530 $\endgroup$ – Christian Nassau Apr 10 '15 at 13:51
  • $\begingroup$ I found this link in Knapp's "Operations and cooperations in Im(J)-theory" which might also be relevant. $\endgroup$ – Christian Nassau Apr 10 '15 at 13:52

This was answered by Don Davis, 1975 Bol. Soc. Mat. Mex. In modern notation, the answer is (at p=2) $H^* j = (A \oplus \Sigma^7 A)/I$, where $I$ is the ideal generated by $Sq^1 \iota_0$, $Sq^2 \iota_0$, $Sq^4\iota_0$, $Sq^8\iota_0 + Sq^1 \iota_7$, $Sq^7\iota_7$, and $(Sq^{(0,1,1)} + Sq^{(4,2)})\iota_7$, in Milnor basis notation. If you prefer the admissable basis, this last operation on $\iota_7$ is $ Sq^6 Sq^3 Sq^1 + Sq^7 Sq^2 Sq^1 + Sq^7 Sq^3 + Sq^8 Sq^2 + Sq^9 Sq^1 + Sq^{10}$.

I derive this quite explicitly and elementarily, as a sample of how to use some sage code Mike Catanzaro wrote, in a talk I gave at Northwestern, "http://math.wayne.edu/~rrb/papers/jnwuhand.pdf", where I also give more precise citation of Don Davis's paper.


  1. you observe that $\psi^3 - 1$ must induce $Sq^4 : H^* \Sigma^4 ksp \to H^* ko$,

  2. you compute the ker and coker

  3. compute that $Ext^1 = F_2$, and that the split extension would give the wrong homotopy,

  4. compute the non-trivial extension using a cocycle defining it.

There, I also note that the Adams spectral sequence for this is quite clean: a marvelous $d_2$ clears out all the rubbish, followed by a simple sequence of differentials which follow from the Leibniz rule, to give the groups of order the 2-adic valuation of 8j in dimensions 8j-1.

Added: I should say, modulo the conjecture that a certain homomorphism really does induce $d_2$, I show this. I do believe it is correct.

Also, there is an interesting module over A(2) that comes up in the j/2 case. The module exhibits periodicity, $\Omega^4 M \simeq \Sigma^{12}M$, and has Ext exactly equal to the associated graded of the homotopy of j/2, in filtrations 2 and up. It is defined over $A(2)$.

  • 1
    $\begingroup$ I am curious about the module $M$. Can you refer me to a more detailed account of $M$ (if it is written down somewhere)? $\endgroup$ – Prasit Apr 12 '15 at 1:44
  • 1
    $\begingroup$ See pp. 38-43 of the notes linked above. It is called F there. I give the periodic resolution, not just a presentation. $\endgroup$ – Robert Bruner Apr 12 '15 at 1:51
  • $\begingroup$ This was computed in MR2171809 Angeltveit, Vigleik ; Rognes, John . Hopf algebra structure on topological Hochschild homology. Algebr. Geom. Topol. 5 (2005), 1223--1290. $\endgroup$ – Robert Bruner Aug 6 at 14:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.